Question

The length of a solid cylindrical cord of elastic material is 32 inches. A circular cross section of the cord has radius 1/2 inch.The cord is stretched lengthwise at a constant rate of 18 inches per minute. Assuming that the cord maintains a cylindrical shape and a constant volume, at what rate is the radius of the cord changing one minute after the stretching begins? Indicate units of measure.

Answer 1

-0..072 Inches Per Minutes

Explanation:

Volume of a Cylinder, V =
`\pi r^2 l`

Given l=32 Inches, r=0.5 Inches

V =
`\pi*0.5^2*32=8\pi \:cubic \:inches`

`(dV)/(dt)=\pi r^2(dl)/(dt) +\pi l(2r)(dr)/(dt)`

Since the Volume is constant,
`(dV)/(dt)=0`

After one minute,

• Length of the Cord=32+18=50 inches

Since the volume is constant

`V=\pi*r^2*50=8\pi\\r^2=(8)/(50) =(4)/(25) \\r=\sqrt{(4)/(25)} =(2)/(5)\\\text{Radius after 1 Minute}=(2)/(5) Inches`

Therefore:

`(dV)/(dt)=\pi r^2(dl)/(dt) +\pi l(2r)(dr)/(dt)\\0=\pi*((2)/(5))^2*18+\pi*50*2((2)/(5))(dr)/(dt)\\0=2.88\pi+40\pi (dr)/(dt)\\40\pi (dr)/(dt)=-2.88\pi\\40 (dr)/(dt)=-2.88\\(dr)/(dt)=-2.88 / 40\\(dr)/(dt)=-0.072 $ Inches per minute`

Therefore, the radius of the cord is reducing at a rate of 0.072 Inches per minute.